Scale Invariant Dynamics of Interfaces and Sheared Solids

In this thesis we study three types of non-equilibrium processes: the depinning of elastic interfaces, the yielding of sheared disordered solids, and the fracture and granular flow of brittle solids. All three display scale-invariant behavior. The first two systems evolve in discrete bursts of motion or avalanches whose magnitudes follow a power law distribution, while breakup of brittle solids produces a power law distribution of grain sizes. The motion of elastic interfaces is studied using simulations of the random field Ising model. The interface is driven by gradually increasing an external magnetic field, leading to a series of avalanches whose maximum size diverges at a critical field. Growth is anisotropic, with the height of an avalanche growing as its width to a power $\chi = 0.85 \pm 0.01$. Scaling relations and finite-size scaling techniques are used to relate $\chi$ to other critical exponents. The roughness exponent of the growing interface is predicted to equal $\chi$ but is substantially smaller at accessible system sizes. Molecular dynamics simulations are used to study critical behavior in slowly sheared disordered solids. The average flow stress rises as a power $\beta$ of the strain rate. Finite-size scaling is used to determine $\beta$ and the exponent describing the divergence of the correlation length with distance to the critical point. The temporal correlations in the average kinetic energy of the system are used to measure the dynamical exponent relating the duration of an avalanche to its spatial size. Lastly, a discrete element model of brittle systems is developed. The model parameters can be calibrated to match specific material properties including elastic constants and fracture toughness. The model is used to study the impact of defect density and strain rate on the fracture of sheared brittle solids. A measure of damage is related to the initial yield and fragmentation. Subsequent granular flow produces a power-law distribution of grain sizes that suggests critical behavior at quasistatic strain rates and large strains. The maximum grain size decreases with increasing strain rate